Strain-gradient elastic-plastic material models and assessment of the higher order boundary conditions

A gradient elastic material model exhibiting gradient kinematic and isotropic hardening is addressed within a thermodynamic framework suitable to cope with nonlocal-type continua. The Clausius–Duhem inequality is used, in conjunction with the concepts of energy residual, insulation condition and locality recovery condition, to derive all the pertinent restrictions upon the constitutive equations, including the PDEs and the related higher order (HO) boundary conditions that govern the gradient material behaviour. Through a suitable limiting procedure, the HO boundary conditions are shown to interpret the action, upon the bodyʹs boundary surface, of idealized extra HO constraints capable to impede the onset of strain as a nonlocality source and to react with a double traction (of dimension moment/area), work-conjugate of the impeded strain. The HO boundary conditions for the internal moving elastic/plastic boundary are also provided. A number of variational principles are proved. A few simple illustrative numerical examples are worked out.